Are {$a {\cdot} b\ \ |\ \ a \in A, b \in B$} and {$a {\cdot} b\ \ |\ \ (a, b) \in A \times B$} the same thing?
Given two sets $A$ and $B$ are the following two sets equivalent?
- {$a {\cdot} b\ \ |\ \ a \in A, b \in B$}, and
- {$a {\cdot} b\ \ |\ \ (a, b) \in A \times B$}
Yes. The Cartesian product of $A$ and $B$ is the set of 2-tupples generated by pairing all elements of $A$ with all elements of $B$. Or
$A\times B = \{ (a,b) | a\in A, b \in B\}$
As there are no restrictions on $A$ and $B$ in your first set it is equivalent to your second set.